so let's go in and start but before I do this anybody have any questions arise so you guys remember last time we started looking at galvanic cells and simply applaud galvanic cells take advantage of the spontaneous redox reactions show willing to take advantage of a spontaneous redox reactions to a flow of electrons through a wire and we looked at the

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experimental setup for the galvanic cell shown here we call this the Daniels cell in the downhill cell is made up of 2 lecturers we have a zinc electrode and copper electoral zinc electrode is the analysts and the cop electrode is the capital and so instead of specifying the lecture you could also use the redox couple right it's like the conjugate acid-base there and always when you list redox couple you must it's so that you have the oxidized for form 1st and then a diagonal line and in the reduced foreign city the redox couple for those intellectual and it will be zinc to plus think if you look at the redox couple for the cop electrodes it would include copper to plants and copper and as all of you know the and nowhere is the site at which the opposition to explain the capital is the site at which reduction takes place and so the oxidation takes place at the end of at the annual is the point at which electrons are lost from the cell because when oxidation takes place electrons are being produced a generated and these are being pushed to the wire are right and the other and we have the capital electrons are being pulled into the cell all right and so the capital is where electrons into the cell I'm the analyst is assigned a negative sign because it's the site at which electrons generated through the wife pushed to the wire and the captain has a plus sign to show that electrons entering the cell immunity would right the to have reactions the oxidation have free actually can see the 2 electrons that produced another reduction have reaction you can see the 2 electrons at take note if you want a record the net reaction in their reaction would be the sum of the 2 and now you can look at the net reaction overall net reaction summarizes what's going on overall this galvanic cell OK and then lastly we look at the fact that we have a shorthand rotational scheme that allows us to summarize his entire picture that drawn up there can be summarized in just using this rotational scheme so what I needed to be able to do so is able to look at picture and write down that the cell notations which describes it all bicyclists and you see a cell notation then would need to be able to do is have a mental picture of what supposed to look like for the setup should look like and how the whole of cell constructed a case usually the fluid look at notations just like what we said when we use advanced equations when we use chemicals symbols they actually represent pictures and macroscopic or microscopic level just as you when you see copper solid you know that means it's metallic copper at a microscopic level you should know that the Giants structure of just copper atoms bonded to each other forming a giant structure right likewise when you see some location at this time he should immediately have a mental picture of what that sell looks like now we finished up within the class looking at some examples of so notations and is going to continue doing that so so that the summarizes what we looked at last time now going to proceed to looking at some other examples of salutations so if you take on the hydrogen electrodes that's right so if I take the hydrogen electrode so overall it's called the hydrogen electrode

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and if you take a height intellectual at the cell notation that describes that is H Class H 2

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so that's the redox couples start this is the redox couple that describes the hydrogen electrode obtained him 100 intellectual it could be a capital is limited it's the capital

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which is the site of the reduction Seoul you would have to to age plus quest plus 2 electrons giving you H 2 guess so because

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the cattle and this would represent a reduction of right and so you would use a cell notation for the cattle parts then remember when you personification you always tell the reacted 1st and then the product so the reactant here is age plus 8 West nite ever-changing thing is you put a vertical line from the ICC was stay with soluble and it was too I guess so there is a change in status so there's going be a vertical line to give you hydrogen gas but as all

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of you recognize for the hydrogen electrode because both 1 is in a close phase and the other is the gasses phase you need some new metal to bring electrons into the South all right Johnson said a commentary in a metal that we use electric chemistries platinum you can use things like carbon fiber and so on and because we have to have them up and your mettle bring electrons and wouldn't hurt that at the end alright and so this would be the notation for the hydrogen electrode if it were capital that I was the right then it turns out the hydrogen is the end of now this is the site of the oxidation so now the reaction would be where electrons of being generated and so you would end up in that and so I'd be an old we have oxidation so now have a right to sell notation for the last because now we will be handled and whiskey and all you start with the inert metal squares and then you write down the reactant which is it's too and then each class requests case of using inert metal to bring electrons into the cell it's in an hour of you always start with that it's a cathode you always end with that so it's kind of both ends of the so that Union of metal that is used to bring electrons and I'm put on the extremes on either side of a case so environment look at the general reaction to remember we looked at 2 types of ourselves last time where we have the hybrid-electric so you take so what I have here the 1st part is electrode where you have a zinc electrode on 1 side that's the an in the hydrogen electrode is the capital with the reduction takes place so if you want to write the net reactions but then you know that the net reaction would be zinc solid class this is the reduction so you're going to have to to age plus was giving you zinc 2 plus 2 equals plus H 2 gas are rights where this is there and of an old wood oxygen takes place as the capital of the reductions he takes place so the cell notation for them when the we start with the an which is zinc and zinc to plus this the reactant product and vertical line of a diagonal line diagonal line is reserved for redox couples attend and then now we have a salt bridge and bring in the salt bridge now that like the captain the reactants for the catalog is H class it quest Whittaker line because you have a change in phase going to H 2 and from the Ganassi ever-changing faces you go to the solid material platinum show and that would be the cells patients when you see the cell notation it translates to the picture that's drawn there OK now advise movement to this cell on side so it might go to this side and look at this hour now you can see but the hydrogen electrode is the and and a couple electrode is the capital of our rights so this is the site of the oxidation so that means the net reaction here would be H 2 gasses are right plus Copper 2 plus 2 equals giving me too h plus 8 plus copper solid OK so this is the reduction have reaction to this is the oxidation have reactions to electrons being transferred all right and therefore the cell notation should be no the hydrogen electrode is the and so we start with platinum then we have hydrogen gas where is the line H plus the quest 2 Whittaker lines but that's all bridge and then on the other side I would end up with the reactor which is copper 2 plus 2 equals Whittaker line copper solid narrative use the 100 electrode depending on whether the hybrid-electric the analyst or the cathode you write some patients will remember you have to keep in mind is all galvanic cells and we only take reactions that are spontaneous so bold these examples that we look at our spontaneous reactions are right and that would be the sole mentation now 1 last thing I want to assure you if we take another example of another electoral tape and let's take the example of the let's say I have this electrode so the redox couple here is I'm going to look at 3 plus line to plus electrodes alright so I have the best then if this weather can lower your memory at the cathode is the site of a reduction so with right the equation in the form of reduction that means you have signed 3 plus Equus plus 1 electron giving you find 2 last request member your have reactions have been balanced and 1 way to quickly check whether your equations balanced is an imbalance 1 9 1 9 9 years have charged balance of 3 plus minus 1 so this is plus on underside therefore the overall chart on that side would be to plus and therefore you have charged balance as well and because electrons of being picked up this is a reduction which alright so you look at that as you have time 3 plus a quest and I'm too plus a course of both of these but neither of these are metals both of these are dissolved ions so commemorate that means if I want this to act as a Category I have something I need something to deliver electrons to the south and so I needed in a metal to deliver deliberate electrons to the south and therefore the cell attention for this would be my reactant which is 3 plus the quest

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OK now if you look at that as there is no change in face the book a question in the same face the fall when you have the same day as the new user commentary indicated in the same thing use of vertical lines if you have 2 different phases of his son used a comedy and then I'll have if the 2 plus 2 equals now I have a change of pace because of the brain the platinum metal and that's where this would look like OK if this were the on all nite you would have the oxidation so this would be the 2 plus equals plus an electron giving me if the 3 plus so this would be the oxidation Nelson since the end of wouldn't start with the inert metal that delivers electrons and then my reactants which would be if the 2 plus 2 equals ,comma 53 plus quest to guys get them because they're both in the same thing is now we use a common yes 0 will you not of you write shoulder a written in the form of reduction thank you to be right you can so that we see a producing electrons and so In 1 way to quickly check whether you made a mistake I'm writing this really fast because but I do want to finish this topic today but 1 way for you to quickly check with that equation is balanced and not used to do in and out charge balance and you'll find that if I had with electrons alongside I'm too plus plus electron would give me plus 1 son Montana plus 1 on the other side plastering their final that the equation is not a cake and this is an oxidation so electrons being produced again now so you undertake take an example but the West if you take the reaction where you have zinc solid plus 2 Of the 3 plus requests because remember each 1 of these produced 2 electrons if this is the reduction in their picking up to buttress of zinc producers to electrons with oxidation takes place and you need to agrees to balance electrons are right because each only if is the IRA is undergoing a reduction in the giant picks up only 1 electron seen twice as many so the balance equation you have to have zinc to plus quest plus 2 if the 2 plus 2 equals case this would be the net reaction that we're looking at and if this were the reactor that we're looking at where zinc is this is the this will be the cathode because this is the reduction to electrons lost I think into electrons of being picked up by a battery multiplying it by 2 questions too OK so that was be the balanced equations and In then buy water right to sell motivation for this might Arnold would be zinc electrode so that the reactant product now I need salt bridge and now my cat load would be where I should have on my react and which is at the 3 plus so have been FEC plus it's worse now ,comma because you both in the same fate if the 2 plus it worse in a vertical line and plaque but it is something a lot of examples to show you how to write a song notations now online homework is going to get you to write so limitations now this online homework will be a little frustrating because you have to use those vertical lines ,comma was so the online homework is that actually checked to see whether you're putting a diagonal line or when collided with you put the ,comma appropriate place alright spacing will become important but it's a good practice and so on and it'll tell you how to put those symbols and will have type it in a paper so please follow the instructions carefully but it's a good exercise because after you find examples I your you could me pretty good at it but you will also find it very frustrating alright because Member computers are not smart it's look to see whether your typing in the correct symbol in the right way take yes the yes you need to indicate the phases are right on an exam because that's how you can tell within the same Faisal at different phases and so on that's how you can tell they even have a Whittaker line over there and have a comma are right yes going 2 you do not include coefficients facilitation alright facilitation is you only identify the species are right the conditions only to balance equations are right you don't even include the coefficients when you look at redox couples redox couples identifies only the reactants and the product the oxidized PCs and the reduced species OK so the alternatives coefficient is where you balance equations and trying to figure out how electrons a given off and how electrons and take the case so I warn you ahead of time giving them ample time when you do that homework celebration because you are going be frustrated because it's all about how you type in trying to typewritten incur correctly he you put an extra space where there should be a space it's going much wrong a and say you know what's going on can also there is a difference between the number 0 and the letter at all a lot of people in this you know that when they write H to all they have you actually put the number 0 because we use in the Geneva font you can't tell the difference between the number 0 hour and The letter all OK and so on make sure you type into parking along and just have whites might you want and the reason is that you're typing in the number 0 rout the letters and vice-versa OK but yes go ahead of me not to I understand yes you do not include if it's a solution unless each tool is part of the reaction where water is being produced to something like that guided it to make solution you do not include the solvent because the solvent is in large amounts it's there in the background and that's what ECU means we put when apprentices ECU that's what that indicates our right and so you do not include the water unless it's part of the balanced equation all right so now that we've looked at cell notation and now we're ready to look at so potential so we're going to look at the cell potential the voltage for Electro motive

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force recalled 3 terms mean the same thing analysts and a symbol that would users in the cell Will they knew the symbol the subscripts and this refers to to the cell potential of voltage or Electro-Motive force non-member whenever we set up a galvanic cell but like the 1 shown here we said that we attachable leader and we measured the voltage so I time equals 0 the incentives started to complete the circuit the voltage that you measure will be the cell potential right and analysts cooked Udall electron start flowing as current starts flowing what happens is all the time you notice that yourself potential gets smaller and smaller so equals 0 as well as some potentially the highest and then after a while as this reaction proceeds and proceeds and proceeds devoted to start decreasing and at some point is going to go to 0 and that means now yourself is spent and that an elderly Gallic so the batteries and that's what happens when you you know with the batteries that we use after a while the batteries gets spent all right and the reason batteries gets spent is because the reaction takes place already badly liked equals 0 the moment to take up the 2 stickers of the red unity to the you know a brand-new Bart the battery you buy it's something you know that a plastic in between there and you have the plastic off and that that is quantity equals 0 it's instead and then what happens is over time the battle gets spent and that means what that means is that equilibrium has been established to begin where you have this taking place in the reactors going to forward direction right so it's like as if you had only reactants no products not not accurate description but I just want to show you that it's going to go take place in the forward direction reaction will proceed in the forward direction on two-week Williams established when he close established now you bet respect our rights and so on so what we measure is called a cell potential act equals the incentive that the reaction starts the voltage that you measure or as reaction continues the voltage that you measure at that time is called a cell potential became so we wanted to find so potential is it's the ability "quotation mark we said if you take galvanic cells galvanic cells push electrons to 1 and so and pushes electrons out the capital pools electrons so you want a kind of have a feel for what it means when we talk about self-protective it's the ability of a cell to push electrons into the wire and to pull electrons arrange the greater ability to push electrons and with the Liverpool electron so that while the higher the cell potential so if you want to to find oneself potential is the seller attention is a measure is a measure but the ability of cell to push and cold electrons through circuit alright so it is the ability to push and pull electrons to a and of course the unit this would be Green units of vaults right now when we take galvanic cells memo were taking advantage of a spontaneous reaction right if the reaction is spontaneous than the reading that you will need a read should always be positive are right and think that in the laboratory do the experiment you connect it to the ball-winner like I said last time it as it is it it's a spontaneous reaction if you bought me reads negative it means that you reversed the polls all right and so you need to be connected so that university and then get a possible for me to read the case so far the fighting the genteel sound for example if I take the Danielle sales this looks that zinc solid going choosing to Class equipment and will using cell notations so wouldn't say copper 2 plus 2 equals of line copper solid that would be the Daniel cell the cell notation is always customary to write down the South potential right next to it so whenever you write a song notation you also will give the cell potential right next to suffered this combination the cell potential is 1 . 1 0 walls right and and when you deal with some potential now it turns out that the cell potential that you measure is related to free energy so when I look at the relationship between cell potential and free Energy Inc and as all of you know for any reaction so the reaction that will looking at how is this reactions of you look at the net reaction we know that for this reaction the symbol for free energies detained infinities delta G and if this is spontaneous we know that the sign has to be negative right if Delta G is negative then your cell potential has to be positive OK that's why did during the galvanic cell he'll have to make sure that yourself pretensions possible guidance and the relationship between the 2 is given by Delta G reaction equals minus and have ye South right so where delta G is the free energy of the reaction and it is the number of moles of electrons transfer In the reaction as given by the balanced

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equation so if you take so if you take the Daniel salad we know that 2 electrons to mold electrons for the balanced equation for every mall of zinc and every Moeller copper that's involved in that we actually you have to let electrons being transferred and therefore and would be to save refers to the number of electrons being transferred as given by the balanced equation OK now there have is called Saturday's constant Farid is constant and various constant gives you the charge per mall of electrons right and this is a constant and this has a value of 9 . 6 8 4 5 times 10 to the power for some qualms from all right with C equals cool on the unit for charge members who wants and so on if you take the fact constant it's a constant value 9 . 6 4 5 4 9 . 6 8 4 5 comes into the powerful qualms from all and what they use the charge from all so we know the number of malls electrons that have been transferred based on the balanced equation we know that each of those electrons will have a charge of 9 . 6 8 4 flights to the 1 mall electrons will have that charge and then of course we know that is the South potential and that's the free energy obtained so it might take the example of the Daniel sell followed you recognize that for the Danielson so let's apply this to the Daniel sell this example is Daniel cell and take Daniel Sullivan net reaction we know it zinc solid plus copper to plus the closed giving using 2 plus 2 equals was copper solid and I know in this environment half reactions I know at be old I have the oxidation which is zinc solid going to zinc 2 plus 2 equals was 2 electrons at the cathode I have copper plus request was 2 electrons giving you copper solid so by looking at the half reactions under 2 electrons being transferred to malls electrons because in a balanced equation and you can see them as long as 1 more of in reacts to the 1 losing 2 plus you have to molds electrons being given and at the capital you can see that 2 malls electrons I taken up so I know that I'm going to use not by 1 account with the free energy for this reaction I know delta G Reaction equals minus and have you sell alright and I know that an equals 2 we know that the cell for them it is 1 . 1 0 volts all right and so have is a constant and I can't think of a delta G 4 that reaction is I say that Delta GE reaction equals negative and I'm going to put the 2 moles times society constant is 9 . 6 5 times 10 to the power for Colom's per mall all right so that's a fairly confident the constant since everything in this is like 2 3 significant figures and is rounding up the constants the 6 things as well and end times the cell potential is 1 . 1 0 balls yes :colon although I may have made a mistake and it should be 9 . 6 4 8 5 would about break down 9 . 6 I got these 2 mixed up because of the 9 . 6 4 8 5 Pirates of the fairly constant should be 9 . 6 4 8 5 OK show around as often you end up with this which gives you so often inexact I will not provide this equation but I will list out all the constant so I will give you what the Friday constant as I write to you need to memorize the product constant but you do need to know how to use that equation you know what equation you need to use it even if figure out so you need to know this relationship OK so you work that out this would come out to be but negative 2 . 1 0 times 10 to the power of 5 right cool on world's from all right now it's important to remember that 1 cool on hold equals 1 Jewel R. I. because remember free energy is always given in terms of killer jewels from all so we have to convert to the SI Unified Energy which is scheduled to kill the jewels are right and the conversion of Kulim holes to jewels is that 1 Colombo is 1 Jewell so that is this equals negative 2 . 1 0 times tends to make attended par-5 Jules Kroll but we always get free energy in terms of killer jewels for more free energy and in improper change change free energy change in and out there is always killer jewels from all entropy change is always wear jewels for Calvin promote case of that that's the sort of standard unit we use so you I can wear them to kill jewels from all and divide by thousand so you end up with negative 210 pound killer jewels from all right and the answer comes out between 6 and figures so let's say you're given an example where you you're asked to give up calculate the free energy for a reaction and let's say it wouldn't apply this to another example of an example that will look at it is let's say for this reaction so we're looking at the reaction of into last 3 each to gas giving you too 3 gas and let's say that the

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cell potential for there is 0 .period although the potential for this is 0 . 0 5 7 balls OK and let's say you're asked to calculate delta G reaction for that OK so that is that if I wanted account what delta G is considered the 1st step I today how many electrons of being transferred all right and so this reaction if you're going to figure out what we've done lots of examples where we figured out how do we know how many electrons are being transferred so if I take this equation and I want to figure out how many electrons of being transferred by How would approach this it's really hard to splits into 2 half reactions to machine that the absence of splits in half reactions What is the way in which we figure electrons have been transferred to balanced redox reactions where we need to do oxidation numbers are right so you can see that I have to nitrogen atoms but each will be 0 because element in its natural state here you can see that I have 6 hydrogen and each hydrogen atom would be 0 0 Over here I have to do nitrogen but now can you figure out what the oxidation number for hydrogen and nitrogen is each hydrogen is plus 3 so each nitrogen has to be worked out negative 3 and then we have 6 hydrogen each would be plus 1 OK so now let's figure out which 1 is undergoing an oxidation said take nitrogen nitrogen goes from 0 to negative 3 as the oxygen of increase or decrease it decreases so you can say that this decreases and therefore it's of reduction are right and each 1 decreases from 0 to negative 3 so what is the relative change 3 so that means it's reduction 3 electrons are picked up each there to nitrogen atoms overall so 6 electrons picked up now that's that hydrogen each hydrogen goes from 0 2 plus 1 so the oxidation of increases in the oxidation number increases we know it's an oxidation Casey verified that oxidation never look at each hydrogen atoms each hydrogen atom goes from 0 to plus 1 so what is the relative change 1 certain oxidation so we know 1 electron is lost Adam how many hydrogen atoms are there 6 2nd you see that 6 electrons are lost have finally you have to conform with the cardinal rule applies you always redox reactions what's the rule number of electrons given enough have people number electrons taken up so that they will each other yet so now we know that for this reaction and we know all that and equals 6 we figured out that 6 electrons of being transferred we know that the equals 0 . 0 5 7 are right so that means Delta G reaction equals miners and have you need salad he wanted to that's negative times 6 times 6 9 . 6 5 times 10 to the fore "quotation mark launched from all times 0 . 0 5 7 bowls and that the comes out to be negative 3 . 3 0 times 10 to the power for cool long holes from all of which is also the same as the . 3 0 at times 10 to the power for jewels from all now I need to convey that to kill jewels divide by thousands you can see comes out at the 33 killer jewels from all I doing this relatively fast because you know we don't problems of this type before where you don't have the energy we've completed the modern hammocks and you know what it means when we look at the free energy economics a galvanic cell delta G has to come out to be negative alright so it's not negative you know there's a problem that OK so now we look at the relationship which we the cell potential and on the free energy now the same relationship if it if the reaction is under standard conditions are right now what we do is we put that will not remember the altered gene on the reaction equals miners and have you not itself alright so if the reactor is taking place under standard conditions then we would write this in this form where United said Is the cell potential under the Standard conditions right and when we say standard conditions for the return of it would really galvanic cells it means that all reactions and products are under standard conditions that means you gasses gasses are at 1 atmospheric pressure solutions are 1 molar right so standard conditions are a standard atmospheric pressure which is 1 atmosphere so it is dealing with galvanic cells and you're dealing with its standard conditions than your reactions and products have to be under standard conditions and that medium gasses involved the pressure the gasses to be 1 atmosphere if you have any solutions involved that means that 1 molar concentrations of copper to class or if the 3 plus it was solution should all be at 1 molar concentration you know what you want finally sort of planning and take-home message to remember is that that whether you're dealing with delta G with it's its understanding conditions or if it's under standard conditions we know delta G is an extensive property write what we mean by what we say Philadelphia quantities like Delta GE Delta H Delta asked extensive property is divided over last quarter when is evening when something is extensive property who were members it depends on the amount of substance remember an extensive quantities like masks if you double the amount the massive doubles already double the volume the volume doubles right and so an extensive quantity is any quite that depends on the amount of substance said you would double the amount they're

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using banned delta G would double if you triple the amount of the deal would triple the same with Delta H or Delta at us but we deal with that potentials so you take sale this is an intensive properties so even if you change the coefficients in a balanced equation double them the cell potential does not change our right so attention quantities like density you take a drop of water the densities 1 the take a whole swimming pool up your water began to be 1 to get it did a bucket of water the densities 1 so density is an intensive property it's independent amount of substance erect likewise when you deal with potential potential it is an intensive properties said you write a balanced equation in you double everything you know all the questions by 2 it does not matter that will not affect the self potential strategic Daniel South and multiply the coefficients for that net reaction you multiply everything by 2 yourself potential remain the same if you multiply the conditions by to your free energy would double all right you indelibly change that reaction would double your interview change that reaction will double right however so potential is an intensive quantity and it does is not affected by changes in the coefficients that equation so that is an important point to remember so now that we've looked at cell potential and the relationship between free energy and some potential now only development this a little deeper and look at what we call standards the reduction of potentials for standard electrode potentials 1st of the 4 the college standard potential all of these terms used to refer to the same thing so it it is standard reduction potential standard electrode potential all the standard potentials OK so it turns out that it is convenient to you take the Daniels cell what we measured experimentally is the potential overall potential when you combine the and would with the capital of and the potential that you measure is the self potential for the entire cell it is convenient for us to split-that that into 2 parts because sometimes it's convenient no what contribution each of those electoral makes and the reason it's convenient is there now if we can assign a value to each of the 2 parts member every gallon itself is made up of 2 electoral you have and and the capital all right and overall which you measured experimentally is the overall cell potential however just as a mental process for us to stick it's convenient to take the XL potential also protested that we measure and splendid 2 parts so that we want to look at what is a contribution that comes from the animals and what is the contribution that comes from the capital is an accessory at nite you in reality you can't measure that that's not experiment measurable but will want to do is willing to sign a relative weight to the contribution that comes from the and a contribution that comes the from the capital of Seoul and defined this term call the standard electrode potential and were going to say it is convenient to think of each electrode as making a characteristic contribution called in its standard reduction potential right said convenient to think of each electrode as making a characteristic contribution call the standard reduction potential as it turns out standard protection potentials Our always written this form of any reduction OK so Will the right everything in the form so we're going to say that each cell has 2 components to it are right and we're going to look at the contribution that comes from each 1 of those and that contribution that comes from each party is called the standard reduction potentials and we by definition we always right the reactions as reduction even by the end of the year an oxidation we write everything as a reduction are right and the symbol for the Is he nodded alright and of course it's a the unit is balls are right the symbol it is not and the unit is In votes so it turns out that you turned the tables you will find tables where all of this is listed out some of these are called so what you will find as you turn the tables you'll find what is known as standard reduction potentials in it was solutions so you find these available in tables alright and steamy 1 moment finished atop OK so this is much like K values remember you have to be a values for assets and all the answers elicited value of thinking about likewise for all we don't have reactions you have standard reduction potentials right and there always will be in the reduction form because if we wanted oxidation all you do is you universe designed so all we can do is another repeat the class next time that I just want to remember that if you're looking at the cell potential best-seller potential is the summer the capital the contribution that comes in the capital Mehmet Kaplan is a reduction plus the contribution that comes from the air and all but now we have to flip the sign because it it's an oxidation remember all these tables are always written in the form of a reduction so we flipped the sign so that now we end up with and all right in this way now we can use

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these tables to add the contribution that comes from the capital of which is the lectured on 1 side and the annual which is the other and something together that you were also attention again so I will would that at the beginning of class time take lots of examples